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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 12705.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.m1 | 12705h2 | \([1, 0, 1, -7505149, -5876790559]\) | \(19827475353801179/5148111413025\) | \(12138977419353046075275\) | \([2]\) | \(887040\) | \(2.9466\) | |
12705.m2 | 12705h1 | \([1, 0, 1, -2653654, 1588689947]\) | \(876440017817099/44659644435\) | \(105305105476389249585\) | \([2]\) | \(443520\) | \(2.6001\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.m have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.m do not have complex multiplication.Modular form 12705.2.a.m
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.